Mathematics

1.GA.1 Transcript

This is Common Core State Standards support video in mathematics. The standard is 1.GA.1. This standard reads: distinguish between defining attributes such as triangles are closed and three-sided versus non-defining attributes, for example, color, orientation, overall size; build and draw shapes to possess defining attributes.

Let’s see what standards are related to this one. If we concentrate on this idea of distinguishing between defining attributes, there was a standard back in kindergarten, K.G.B.4, that speaks to analyzing and comparing two- and three-dimensional shapes in different sizes and orientations, using informal language to describe their similarities, differences, parts, and so forth. Also at the kindergarten level, there was standard K.G.A.2, which states: correctly name shapes regardless of their orientations or overall size. Another related standard in kindergarten was K.G.B.5. This standard is about modeling shapes in the world by building shapes from components. Yet another kindergarten standard related to this first grade standard speaks to composing simple shapes to form larger shapes, and this was standard K.G.B.6.

Looking ahead, in second grade, standard 2.G.A.1 is related to this one. It talks about recognizing and drawing shapes having specified attributes, such as a given number of angles or a given number of equal faces. The second grade standard does get a little bit more advanced because it does talk about angles, faces, quadrilaterals, and pentagons, some terms that are not used in first grade.

One question with this standard would be, well, what shapes are we dealing with here? Back in kindergarten, in the introduction, there’s this statement that talks about identifying, naming, and describing basic two-dimensional shapes, and these shapes are squares, triangles, circles, rectangles, and hexagons. And it also talks about some three-dimensional shapes, those being cubes, cones, cylinders, and spheres. So, that gives us an idea of what shapes we should deal with here.

When we consider the shapes, drawing a two-dimensional representation of a three-dimensional figure is difficult, and that’s probably beyond the scope of first grade. The concepts of cones, cylinders, and spheres do not appear in the standards again until the eighth grade, which is a little bit strange, but that’s how the standards are laid out. I’m sure that we will be addressing those many times before eighth grade, but that’s the next time that these three-dimensional shapes are mentioned in the standards. Also, note that rectangular prisms are not mentioned until the fifth grade. Considering what shapes to use, there’s only one in this standard 2.G.A.1 that is three-dimensional and that is cubes, but, being that we’re focusing, of course, on 1.G.A.1, the focus does need to be on two-dimensional figures.

Here’s a bit of a warning. Most of the manipulatives that you might use will look something like this. For example, this one is a square, but when you consider that original drawing versus the second, here’s the huge difference. The square does not include all that shaded area inside. That’s the interior of the square in this case. Keep in mind that the square is actually just those four segments. So, it’s important that students realize this and that you point that out when you’re dealing with these physical representations.

Let’s focus on this last part of the standard–build and draw shapes to possess defining attributes. So, what about these defining attributes? Well, let’s start off with giving the students three segments. They’ll take those three segments, and of course, the resulting shape will be a triangle, and they’ll also notice that it’s composed of three segments. Another conclusion is that it has three vertices, and another conclusion would be that it is a closed figure. Some students could take those same exact three segments and build a triangle out of it. Now, even though they use the same three segment—they came up with these triangles–and if you have them compare and do something like this, they’ll realize, hmm, given those same three segments, we all built the exact same triangle. So this is an opportunity to lay the informal foundation for the fact that only one triangle can be constructed from three given lengths.

Let’s say you give the students these three segments. Okay, let’s take the longer one and then another one, then another one. Something interesting is happening here. I can’t seem to be able to make a triangle out of this. Let’s say these were the measurements: 4, 2 and 8. Even though the standard doesn’t talk about this, this is an opportunity to lay the informal foundation for the fact that the sum of the two shorter segments of a triangle has to be more than the longest segment. Otherwise, you can’t make a triangle out of it.

Let’s say the students are now given four segments with two being equal lengths. Okay, let’s do this; let’s take this one, then a second, then a third, then a fourth. Then we put them together. Conclusions here would be that this figure is composed of four segments; it is closed; it has four vertices. What students will notice also is that a four-sided figure doesn’t necessarily have to be a rectangle or a square. It can be something like this. Let’s say you give the students four segments that are all equal lengths. Well, let’s see. What can we make out of that? Well, some students might make this figure, which is a square. Other students might take those same four segments and do this. Unlike triangles, different figures can be made from the same four line segments.

Let’s look at this idea of an angle. Although the concept of an angle is not introduced in the standards until Grade 2, the foundational knowledge can be explored at this level because it is a defining attribute. So let’s take these two segments. We can join them together at one vertex to make this size angle, and that angle could be part of, let’s say, a figure like this kite. Let’s say we take those same two segments and we have a larger angle, and in fact, it’s a right angle that could be the angles that compose a rectangle. Then we have a much larger angle here, which again could comprise part of the angles in a quadrilateral such as this one.

Students should notice a pattern that if you change the size of an angle, you will cause a change in the shape. For example, let’s say we start off with these two segments, and we have this size angle; and we can make this triangle out of it. But then we could take those same two segments and make a smaller angle, and when we connect those two endpoints, we have a different triangle because we changed the size of the angle even though we started off with the same two segments as before. And a third example, we’re still using those same two segments, but we made the angle smaller. Now we connect those two, and we have yet another different triangle. So even though the term angle isn’t used at this grade level, it is important that you somehow informally introduce it because it is a defining attribute. It will change the shape of an object.

There are some shape patterns that emerge when using segments of equal lengths. So, for example, if you take three equal lengths, you are going to form this type of triangle. If you take four equal lengths, well, you can have either a square or this other figure that we call a rhombus. Students will get this type of pattern when you combine six equal-sized segments. Although it’s not mentioned at this level, students could construct a pentagon using segments of equal lengths. So, you could give them five equal-sized segments and have them make a pentagon and they’ll see a pattern as far as having this same exact shape every time.

Let’s look at the non-defining attributes. Color is a non-defining attribute. I’ve got this figure, and I’ve got this second figure. They’re the exact same size and shape and everything. They’re different colors, but again, that’s not a defining attribute of these. Overall size is also a non-defining attribute. Figures can be the same shape but not necessarily the same size such as these two examples. Here are additional examples. We have these two rectangles. Again, they’re the same shape, but they’re not the same size. Orientation is also a non-defining attribute. Changing the orientation does not change the original triangle into a different one. So if we start off with this triangle, we changed the orientation a couple of times, but we still have the same triangle. The triangle itself did not change. It looks different, but it’s still the same triangle. Same thing with this right triangle—we started off with this original and we changed the position a couple of times, but we didn’t change the triangle itself. It’s still the same one.

We can do the same thing with this quadrilateral. Again, it looks different when we change the orientation, but it’s still the same quadrilateral. If you look at these triangles, they’re different colors. They’re different sizes and orientations, but they’re all still triangles. Informally these are also different shapes, but they’re still triangles. So, let’s look at this word, shapes. The glossary in the standards does not define shape or figure. Shape typically refers to the form or outline of an object. In mathematics, figure can refer to numerals, but of course, a figure in math can also refer to a geometric form such as a square, triangle, or sphere.

What about this idea of similar shape versus similar figure? If we look at these two triangles, well, these are actually similar figures. Maybe we should get a little bit more formal here and think figures rather than shapes. For example here, look at all these different types of geometric figures. Mathematically, these are all different shapes, but they’re all four-sided figures. Their shape appears different and everything, and the shape is different in each one, but they still have the common aspect that they’re all four-sided figures. So, they’re different shapes, but they are the same type of figure because they’re all quadrilaterals. If you stop and think about it, the different shapes occur because of the differences in the sizes of the angles and the lengths of the segments in all of these four-sided figures. If we change the number of segments, in this case to six, we have all these hexagons. Again, the different shapes occur because of the differences in the sizes of the angles and the lengths of the segments, but they’re all still hexagons.

Let’s conclude by looking at the standards for mathematical practice. If we look at the first four of those standards and we did the activities that were involved in this standard, students would reason abstractly and quantitatively. They would construct viable arguments and critique the reasoning of others, and they would model with mathematics. If we look at the last four of the standards for mathematical practice, students do attend to precision and they do look for and express regularity in repeated reasoning.